Bertalanffy Ludwig Von 1957 Quantitative Laws in Metabolism and Growth

8/22/2019 Bertalanffy Ludwig Von 1957 Quantitative Laws in Metabolism and Growth

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Quantitative Laws in Metabolism and GrowthAuthor(s): Ludwig von BertalanffyReviewed work(s):Source: The Quarterly Review of Biology, Vol. 32, No. 3 (Sep., 1957), pp. 217-231Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/2815257 .

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VOL. 32., NO. 3 September,957THE QUARTERLY REVIEWofBIOLOGY

QUANTITATIVE LAWS IN METABOLISM AND GROWTHBYLUDWIGVONBERTALANFFYBiological esearch, t. SinaiHospital, ndUniversityf outhernalifornia,osAngeles

INTRODUCTIONTHE workreviewed in this paper saimed at establishing onnections e-tween wofundamentalspectsof ivingorganisms,heirmetabolismndgrowth.What we call growth f even a simpleorganism s a tremendouslyomplex henomenonfromthe biochemical, hysiological, ytological,and morphologicaliewpoints. here re,however,certain spectsthatare amenable o quantitativeanalysis, nd such an approachappears to leadto some insight nto the connectionsbetweenmetabolism nd growth, nd to some answer totheseeminglyrivial, ut in facthardly xploredquestion,"Why does an organismgrowat all,and why, aftera certain time,does its growthcome toa stop?"QUANTITATIVE RELATIONS BETWEEN BODY

SIZE AND METABOLIC RATEIn order o begin this nvestigation,t maybeemphasized hat, nmanyphysiologicalctivities,theabsolute ize ofthebody s a most mportantfactor etermininghe rateofprocesses.Whetherwe take total metabolism, eart or respiratoryrate, the chemicalcomposition f the organism,excretion,r theenzyme ontent fthecells-wealwayswillfind hattheycharacteristicallyarywith body size, thisbeingtrueeven though hetheorganismsompared n suchrespects howatremendousiversityntheir natomy, hysiologi-cal mechanisms, daptationsto certain environ-ments, nd so forth cf. Adolph,1949). To give

just one example:pulse rate in mammals loselycorrespondso the Y powerofbody weight verseven ordersof magnitude,froma dwarfbatweighing ome four grams, to an elephant of2000 kilograms,n spite of the fact that theanimals under comparison belong to differentorders, nd are adapted to all sorts of climateandwaysof iving Fig. 1).The relationbetweenmetabolic ate and bodysize belongsto theclassical topicsofphysiology.It goes back over morethana hundredyears tothe timewhenSarrusand Rameaux, Bergmannand Leuckart,and Richet noticed that weight-specificmetabolicrate, that is, the intensity fmetabolisms measuredby oxygen onsumptionorcalorieproductioner kilogramfbody weight,decreaseswithincreasing ody size. A classicalexample is provided by Rubner's experimentswithdogs of differentize (Table 1). It appears

that metabolicrate per kilogramdecreases. f,however, metabolism s calculated per unit ofbody surface, approximatelyconstant valuesappear. The comparisonof metabolic rates inmammals led Rubner to the contentionthatwarm-blooded animals produce daily about1000 Cal. per squaremeter fbody surface.Thisis the originof the famoussurface ule, whichwasexplained yRubnernterms fhomeothermy.All warm-blooded nimalsheat theirbodies to atemperaturefca. 37?C. Heat outputtakesplacethroughhebody urface. ence,the amenumberof caloriesmustbe producedper unit surface norder omaintain hebody temperatureonstant.There are, however,considerabledifficulties217

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218 THE QUARTERLY REVIEW OF BIOLOGY1000

IL L co PD o ED6Eh'0G RABBIT~~~~80O OVS~~~~~~~MQMTO-'

= 3 X - - -~~~~~~~~~~~~~_TA SS XctS0O C 4W" I ,UOSELjEP 200O kg.39 5Sg log !50Q ioOs 509 g kg SW tk!g 50k(s l0kg 500kq IOQOV9BODY WEIGHT

FIG. 1. ALLomETRic DZEPENDENCE OF PULS5E FEREQuENcYONBODY WEIGHT N MAMMLSIt may be assumed that the volume of blood transported per minute is proportional to the basal metabolicrate, as the oxygen consumed must be transported by the blood. This volume is equal to stroke volume (S) X

pulse frequency (F). In a rough firstapproximation, S may be taken as proportional to body weight (W). Thebasal metabolic rate follows interspecifically, n the series ofmammals, the W! rule. Hence:S.F. = CW.',and= = C'W

The igmre hows that the allometry onstantofpulse frequency, =-.28. Notwithstandinghegrossoversimpliricationhichneglects natomical,physiological,cologicalond otherdifferences bsolutebodysize is thedominating actor n the control fpulsefrequency,n a rangefrom he dwarfbat (4 g. body weight)to theele-phant (2000 kg.). Modified fterBertalanffy1951a).TABLE 1Metabolism n dogsAfterRubner 1902).

Weightnkg. Cal. production Cal. productionerperkg. sq. m.bodyurface3.1 85.8 19096.5 61.2 107311.0 57.3 119117.7 45.3 104719.2 44.6 114123.7 40.2 1082

30.4 34.8 984in measuring the outer surfaces of animalsexactly, ut a simplemathematical evice can beapplied. If two bodiesare reasonably imilar nshape, their surfaces an be expressed s a 23power fweight,ince hecubicrootofthevolumeorweight s a lineardimension,nd thereforetssquare has the dimensionfa surface.Hence,thesurface reas of geometricallyimilarbodies canbe obtained by multiplying he23 power of theweight y a suitableconstant. his is seen n thewell-knownormula fMeeh:

S =bW (1)

The surface rule of metabolismaccordinglystates that the basal metabolicrate is propor-tional to the/%powerof theweight.n thecaseof man, the determinationf the basal metab-olism is a clinicalroutine, n orderto diagnosethyroid isorders nd the ike.Here thesomewhatmore complicatedDubois formula s applied.Dimensionally, owever, he Dubois formula sidentical with the surface rule. The Duboisformulas: S = kW0 25 L0725.As,presupposinggeometricalimilarity,engthL = cW', thiscanbe written: = kWO-420.W0 725(0 3) = bWThe relationbetweenmetabolic ate and bodysize can be studied itherntraspecifically,.e.,bycomparingnimals f he ame pecies nddifferentbody size, or interspecifically,.e., by comparingadult animals of differentpecies. We are atpresent mainly concerned with intraspecificcomparison.A grave objectioncan be raised against thesurface uleas found n textbooks fphysiology.In consideringhe quantitative elationbetweenmetabolic rate and body size, homeothermicvertebratesand, in particular,mammals arealmost solely taken into account (e.g., Brody,1945; Kleiber, 1947; Krebs, 1950). However,thecase ofmammals s byno means imple ut rather



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QUANTITATIVE LAWS IN METABOLISM AND GROWTH 219is intricate.Moreover, s we shall see presently,many familiar conclusions and explanatoryhypothesesall flat f not onlymammalsbut alsopoikilothermicertebrates nd invertebratesretaken ntoconsideration.t is thereforeecessaryto consider heproblemon the broaderbasis ofcomparativephysiology.A considerable art ofthisworkhasbeen carried hroughn the author'slaboratories.In order o understand hese results, ne moremathematical ormulas necessary.The depend-ence of themetabolic ate of an animalon bodysizecan be expressedn theequation:

M = bWG, (2)where M is the metabolic rate per unit time,W the body weight, nd a and b are constants.This is a special case of the so-calledallometricfornula (Huxley, 1932) which expresses thedependence n bodysize for n enormousmountofmorphological,iochemical, hysiological,ndevolutionary ata. This formula an further ewrittenn thefollowing ay:

logM = logb +? logW (3)That is to say, if metabolicrate is plotted

against body weightdouble-logarithmically,eobtaina straightinetheslopeof which ndicatesthe onstant. If = 3, themetabolic atefollowsthesurface ule. fa = 1 or theslopeis 450, themetabolicrate is proportional o weight.With1 > a > 23, an intermediaryase obtains.If weight-specificalues are taken,that is, ifmetabolic ate per unitweight s plotted nsteadof that of the totalanimal, heequationbecomes:M =bW (4)

Correspondingly,eight-specificetabolic ates,as a general ule,decreasewith ncreasing eight,and theslopeofthe ogarithmiclot s negative.Afterthese preliminaries, e can summarizethe experimental esults in the followingway(Bertalanffy,941b, t seq.):1. The surface ule also holdsforpoikilothermicvertebratesnd certain nvertebrates.he rule s,therefore,fa wideapplication;but theexplana-tion given by Rubner is too restricted, or inpoikilothermicnimalsthere s no thermoregula-tion, and thus the latter cannot be the basicfactor in the relation between body size andmetabolic ate.2. On the otherhand, there are many classes

TABLE 2CO2 roductionfArmadillidiumallasii(Temperature1?C.)AfterMUller 1943b).Weightnmg. 15 33 50 100 160Cmm.C02/hr. 3.0 5.2 7.2 11.2 15.2Perg./hr. 200 174 144 112 94Perunit urfaceWI)/ 48. 5 54.2 53.0 49.8 51.6hr.of animals in which the surfacerule does nothold.3. Thus we come to thestatementhat severalmetabolic ypesexistwith respect o the relationbetweenmetabolic ate and bodysize.In view of what was said previously, hreemetabolicypes, hat is, threedifferent ays ofdependenceof the metabolicrate on body sizecan be distinguished,hisclassificationpplying,as was emphasized,to intraspecific llometry,that is, to individualsof differentizes or togrowingnimalswithin nespecies.In thefirst ype,metabolic ate is proportionalto a surface rthe23 power f theweight.Repre-sentativesof this type include fishesbut alsocertain nvertebrates,uch as crustaceans, lams,and ascaris. Table 2 presentsone example,themetabolic rate in the sowbug, Armadillidium.As can be seen, ts oxygen onsumption er unitweightdecreaseswithincreasing ody size, butremains onstantper unit surface.Subsequentlyit will be seen that sowbugand companyrevealquite a bit about human growth s a centralproblem fphysiology.The secondtype s quite different. ere themetabolic rate is proportionalnot to surfacearea,but to weight tself, o oxygen onsumptionin an animalofdoublesize is simplydoubled, nan animalfour imes s large s quadrupled, tc.Directproportionalityfmetabolic ate toweightis found n growingnsect larvae and hemime-tabolous insects,as well as interspecifically,ncomparing magos of different elated species.Table 3 shows metabolicrates in the walkingstick, Dixippus morosus.Oxygen consumptionper gram nd hourappearsto be constant ver awiderange, overing ll bodysizesand the entiredevelopment.Other groups belongingto thistype are land snails of the familyHelicidae,intraspecificallys well as interspecifically,ndannelids uchas theearthworm.Finally,in the third ypemetabolicrates are



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220 THIEQUARTERLYREVIEW OF BIOLOGYTABLE 3OxygenonsumptionfDixippusmorosus(Temperature0?C.)After uller1943a).

Weightnmg. 8 130 250 450 630 850Cmm. 2/hr. 2.0 30.6 60.7 113.2 154.8206.6Per g./hr. 250 236 243 252 245 242TABLE 4OxygenonsumptionfPlanorbisp.(Temperature3?C.)After ertalanffynd Muller 1943).

Weightnmg. 30-35 8-62 0-100 40 190-200Cmm.02/hr. 2.3 3.9 5.4 7.3 9.5Per g./hr. 69 65 56 52 48Per unit surface 22.9 25.1 26.1 27.0 28.2(WI)/hr.intermediate etweenproportionalityo weightand proportionalityo surface rea. To thistypebelong uchpondsnails s Planorbis ndLymnaeaand filatwormsike Planaria. Table 4 gives datafor heramshornnail, nd shows hat tsmetabolicrate decreases with respectto its weight, butincreaseswith espect o itssurface rea.The relationsmentioned retypical nd charac-teristic f thespeciesconcerned. able 5 givesasurveyof available observations.A few minordiscrepancies eed elucidation, ut in general tcan be said that the "metabolictype," i.e., therelation fmetabolic atetobody ize, s a physio-logical characteristic f the species or groupofspecies oncerned.

INTERPRETATIONS OF THE SIZEDEPENDENCE OF METABOLISM

We now come to the question,what is at thebasis of the relationbetweenmetabolicrate andbodysize and, in particular, f its mostfamiliarform, hesurface ule? Cf. Kleiber, 1947;Berta-lanffy, 951a; Bertalanffynd Pirozynski, 953;withfurthereferences).We must admitthat wedo notknow.What can be shown, owever,s thattheexplanations suallygiven re insufficient.Thereseems obe, first,he alternativewhetherthedependence fmetabolic ateon bodysize isbased upon cellular or upon organismic factors.That is tosay,thedecrease fmetabolic ate withincreasingbody size may be due to intrinsicdifferencesnthemetabolismfthecellsof maller

and larger ndividuals, n whichcase it shouldalso be found n the respiration f tissuestakenout of theorganism; r else, t maybe regulatedbyfactors resent nd activeonly n theorganismas a whole. Let us start with the organismichypotheses.The mostfamiliar ne has already been men-tioned, namely, thermoregulation.here is nodoubt that energy xpense forthermoregulationforms considerable artofthetotalmetabolismin homeothermicnimals.However, hisexplana-tion cannot be general ince thesurface ulealsoapplies, nd infact s more ccurately stablished,in cold-blooded ertebrates nd even in certaininvertebrates here here s no thermoregulation.

Another nterpretationssumes that the sur-facerule s basedupon theanatomyndphysiologyof thecirculatoryystem. he supply of oxygenand nutritivematerials o thetissues s naturallya function f the intensity f the blood current.The latterdependson factors uchas thesize andstroke volume of the heart, the frequencyofheartbeat, thediameter f thebloodvessels, hedegree of capillarization, nd the like. As hasalready been indicated,there are ratherstrictquantitativerelationsbetweenbody size, meta-bolic rate, and pulse rate. Thus, in interspecificcomparison fromthe mouse to the elephant",pulse rate decreasesapproximately roportionalto the % powerof theweight Fig. 1). So doesbasal metabolic ate nthe nterspecificomparisonofmammals,f adult specimens fcorrespondingspeciesare plotted Brody,1945; Kleiber,1947).However,hemodynamicsannot offer generalexplanation.Remember, orexample,the clams,where the circulatorysystem is completelydifferentrom hat found n vertebrates,r recallascaris,whichhas no blood circulation t all-animalswhosemetabolic ateneverthelessollowsthe surface ule.Recent nvestigationsftheLudwig aboratory(Ludwig,1956; Kienleand Ludwig,1956; Sattel,1956) give some supportto the hypothesis ro-posed by Ludwigand by Bertalanffy1951a, p.252f.) that the "metabolic ypes" are connectedwithtypes frespiratorypparatus.Gill-breathinganimals ppearto followhesurface ule;hence tsvalidity n fish nd certain nanimate lasses.Onthe otherhand,the surface f tracheas n insectlarvae developsproportionalo bodyvolume, swas shown ySattel 1956) inBombyxmori;hencethe proportionalityf metabolicrate to weight.



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TABLE 5Relationbetween etabolic ate nd body izeRespiration roportionaloSpecies Reference W2'8surface),W (weight)or intermediate

PLATYHELMINTHESDugesia gonocephala Bertalanffynd Muller, 1943 IntermediateNEMATHELMINTHESAscaris umbricoides Kruger,1940 SurfaceANNELIDALumbricusp. Muller,1943b WeightEiseniafoetida Kruger,1952 Surface?MOLLUSCALamellibranchiataAnodonta ygnaea Weinland,1919 SurfaceDreissensiapolymorpha Ludwigand Krywienczyk, 950 SurfaceProsobranchiaLithoglyphus, aludina fasciata andP. vivipara Krywienczyk,952a SurfacePulmonataLymnaea tagnalis Bertalanffynd Muller,1943 IntermediateLymnaea tagnalis Fusser and Kruger,1951 IntermediateLymnaea curicularia Krywienczyk,952b Weight?Planorbis p. Bertalanffynd Maller, 1943 IntermediatePlanorbis orneus Fusser and Kruger,1951 IntermediatePlanorbis orneus Krywienczyk,952b IntermediateIsidora proteus Krywienczyk,952b IntermediatePulmonata and Operculata,15species v. Brand,Nolan and Mann, 1948 Surface (high ten-intra-andnterspecific perature 30?C.]?)HelicidaeHelix, Chilotrema,nd Cepaea (inter- Liebsch,1929 Weightspecific)Cepaeavindobonensis Bertalanffynd Muller,1943 WeightCRUSTACEABranchiopodaDaphnia pulex Jan6aroik,948 SurfaceArtemia alina Bertalanffynd Krywienczyk, 953 SurfaceIsopodaAsellusaquaticus Muller,1943b SurfaceAsellusaquaticus Will, 1952 SurfaceArmadillidiumallasi Muller,1943b SurfacePorcellio caber Will,1952 IntermediateOniscus sellus Will,1952 SurfaceLigia oceanica Ellenby,1951 Probablysurface

DecapodaAstacus astacus Kalmus, 1930 Surface?Potamobius orrentiuin Wolsky,1934 Weight?Pugettiaproducta Weymouth t al., 1944 IntermediateHomarusvulgaris Thomas,1954 SurfaceINSECTAHemimetabolaDixippus morosus Muller,1943a WeightHolometabolaVariousspecies, ntra- nd interspe- Kittel,1941 WeightcificTenebriomolitor Bertalanffynd Muller,1943 WeightPISCESLebistes eticulatus Bertalanffynd Muller,1943 SurfaceVarious species (Scorpaena,Abramis, Jost,1928 SurfaceCyprinus, tc.)REPTILIALacerta Kramer,1934 Surface

221



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222 THE QUARTERLY REVIEW OF BIOLOGYIntermediateaseswouldresult rom he presenceof twotypesofrespiratorypparatus.Still another xplanation f the surface ule isbased upon anatomicalor chemicalchanges ncomposition ith ncreasingodysize. "Metaboli-callyactive" organs uchas theviscera, hebrain,etc.,are relativelyarger n smallas compared olarge animals. So it can be assumed that theyconsumerelativelymoreoxygen nd are respon-sible for the higher weight-specific etabolicrate in smallerorganisms. owever,the relativegrowth f nner rgans s verydifferentrom neorganto theother, nd so it is improbablehat tcan yield the simplerelationof the surface uleofmetabolismcf.Bertalanffy,951a).A quantita-tive estimate Bertalanffynd Pirozynski, 953)showsthatthisfactors notsufficiento accountfor heactualvariations f basal metabolic ate.Nowwe cometo the nterpretationsn terms fintracellularactors. his amounts o saying hatthe decrease of weight-specific etabolic ratewithincreasing ize, as expressedn the surfacerule, is due to a correspondingecrease in therespirationf issues. issuerespirationsmeasuredas Qo,, that is, ,l 02/mg.dry weight/hr.,sdeterminedwith the Warburg apparatus. Aconsiderable mountof workhas recently eendone along these ines, partly timulated y ourownworknow to be presented,ust as we mayalso say that the interestn comparativemetab-olismas classifiedn the metabolictypesmen-tionedhas been stimulated y the investigationsongrowthawsto beexplained ereafter.heques-tionof he sizedependencef issuerespirationsacontroversialne, but the statements o followappeartobe a fair resentationfthecase.In interspecific omparison of mammalianspeciesofdifferentizes,ranging rom hemouseto the horse,a decreaseof Qo, with increasingbody size is generallyfound,as a numberofobservershave established Kleiber,1941; Wey-mouth,Field, and Kleiber 1942; Krebs, 1950;Martin and Fuhrmann,1955). This decrease,however,s notparallel n the variousorgans nd,as a general ule, s less than wouldcorrespondothe urface uleorthe3 power ule fmetabolism.In a correspondingay, decreasewith ncreasingbody size was found n enzymatic ystems on-nectedwithrespiration,uchas in theconcentra-tion of glutathione Gregoryand Goss, 1933;Patru'sev, 937),ofcytochrome (Rosenthal ndDrabkin, 1943), of cytochromexidase (Kunkel

and Campbell, 1952), of succinodehydrase ndmalicodehydraseFried and Tipton,1953).The picture,however, s differentn intra-specific omparisons,s betweenrat tissuesfromanimals of different ody size and age. Sevenmain organs f the rat have been nvestigated yBertalanffynd Pirozynski 1951, 1953), andskeletalmusculature, hich s particularlympor-tantbecause it forms highpercentage f bodymass, was studied by Bertalanffy nd Estwick(1953). As Fig. 2 illustrates, o significanteclineofaverageQo2with ncreasing odysize is foundin brain, lung, and kidney; a slight decline inskeletalmuscle, iver,and heart; and a markeddecline n thediaphragm. o theres nosystematicdecreaseof Qo, in the variousorgansconsistentwith, and responsiblefor the decrease of theweight-specific etabolic rate with increasingbody size. These resultshave been essentiallyconfirmed y other workers and with othermaterials: n growing hicken by Crandall andSmith 1952), in the heart muscleof the guineapig by WollenbergerndJehl 1952), ntheteleostbrainby Vernberg nd Gray (1953), and in rattestes by Homma (1953). Similarly, ried andTipton (1953) did not find a decrease in thecontent frespiratorynzymes.From this it would appear that genetic, ndhence pecies-specific,ifferencesn theenzymaticactivityndQo,oftissues re foundn interspecificcomparisons. ifferences,owever,n animalsofthe samespecies nd differenteight re rregularwithrespect o thevarious issues r areabsent.So we have to assume factorswhich,withinthe intact organism, egulatethe respiration fthetissues, hesum total ofwhich s themetab-olism of the entire animal,but whichdo notshowup in the isolated tissueused forWarburgdeterminationBertalanffynd Pirozynski, 951,Schmidt-Nielsen,Bertalanffy, nd Pirozynski,1951). We have already said that the organismicfactors usually contemplateddo not offerasatisfactoryxplanation.What one may expectcan be illustrated y theaction ofthyroxin.t iseasy to induce an increase of metabolismbyinjection fthyroxinntothe animal n vivo; butin spite ofmanyeffortsmade,nobodyhas beenable to reproduce atisfactorilyhiseffect y anadministrationf thyroxino a tissue n vitro.On the otherhand, chronichormonal onditionsare manifestedby significant hanges of thetissueQo, as has been shownwith tissuesfrom



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QUANTITATIVE LAWS IN METABOLISM AND GROWTH 223

15

7.~~~~~~~~~~~~~~z6 t--Y?US

431 Jl_a 1o 20 30 40 50 60 eo MO no 300 400BODYWEi6rTING.FIG. 2. TIssuE RESPIRATIONr VARious ORGANS F TIE WHITE RAT IN RELATION OBODYWEIGHT

Qo2 = .dl 2/mg.drywt./hr.Only regressionines are shown;for ndividualdata and statisticalevaluation cf. the originalpaper. AfterBertalanffynd Pirozynski1953).

hypophysectomizednimals and in pituitarydwarfmice,whichacksomatotrophinBertalanffyandEstwick,1954).The writer oesnot feelhappyabout this tateofaffairs,nd thesituationwouldbe muchmoresatisfactoryfa straightforwardelationbetweenthe decrease of the weight-specific etabolicrate and the tissuerespiration ould be found.Indeed, the Ottawa studywas startedwiththisexpectation,which,unfortunately,as not borneoutbythe facts.The explanation f the surface ule and of thesize-dependence f metabolism n general thusremains ather nsatisfactory.ehave, tpresent,to take themetabolic ype, n the sensedefined,as an empirical atumof the speciesconcerned.However, even this cautious attitude leads tocertainremarkablenferences ithrespect o theproblem fgrowth.

METABOLIC TYPES AND GROWTH TYPESIt has already been stated that, among thevarious animalclasses, so-calledmetabolic ypescan be distinguished y virtue of the relation

betweenthe metabolic rate and the body size.Now as there re different etabolic ypes, hereare also differentrowthypeswhich are distin-guishedby the courseofgrowth s expressed n

thegrowthurves fthe everal pecies. t appearsthat we have been successful n establishingdefinite nd strict onnection etweenmetabolictypes and growthtypes, in consequence of ageneral heoryfgrowth hich stablishesationalquantitative aws of growthand indicates thephysiologicalmechanismupon whichgrowth sbased.Let us startwith rather bviousdeliberation,first ndicatedby Putter (1920). Animalgrowthcan be considered resultof a counteractionfsynthesis nd destruction,f the anabolismandcatabolism fthebuildingmaterials fthebody.There will be growth o longas building p pre-vails overbreaking own;theorganism eachessteady state if and when both processes areequal. We mayexpress his n a general ormula:dWIdt = nWWmKWn. (5)

In words: The change of body weightW isgivenby the differenceetween heprocesses fbuildingup and breakingdown; v and K areconstantsof anabolismand catabolismrespec-tively, nd the exponentsm and n indicatethatthe latterare proportionalo somepowerof thebodyweightW.Obviouslythe growthof any organism s ofan enormous omplexity, hetherwe consider t



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224 THE QUARTERLY REVIEW OF BIOLOGYfrom biochemical, hysiological,morphological,or any other spect.However, he ustificationfan overall ormula nd the implemodel t impliesliesin the following. ur equation tatesthat thegross esult fsyntheticnddegradative rocesseswithin heorganism ollows he aw of allometry,that is, that the rate of theseprocesses an beexpressedas a power function f body mass.But thisassumptions justified, ecause at leastin a firstpproximationherateof ll physiologicalprocesses ithertonvestigatedanbe expressednallometricor power formulas Adolph, 1949).The intrinsiccomplexityof the phenomenonconcerneddoes not precludeit fromfollowingsuch simple, eneralaw.Remember,or xample,what has been found n the dependence f thebasal or restingmetabolism f theintactanimal.Ofcourse,what s calledthebasal metabolic ateis, in fact,theoutcomeof innumerablend to alarge extent unknownprocessesof intermediarymetabolism.Not only this, but the growingorganism ndergoes hangesat the biochemical,physiological,ellular, nd morphologicalevels.Nevertheless, e can statequitedefinitelyhatacertainorganism beys, let us say, the surfacerule; thatis, that therate ofmetabolism f theentire nimal,whateverts size or developmentalage, can be expressed s a function f the -'power f tsrespective odyweight.Wehave nowmore losely odefine heprocessesappearing n our basic equation. The catabolicprocessesmean,ofcourse, hecontinuous oss ofbuildingmaterial s it takesplace in any livingorganism. iochemically,his meansthe turnoverofbuildingmaterials nd particularlyfproteins,as demonstratedby the isotope techniques.Cytologically,t means the renewalof cells, asfound n many tissuesand organs,oftenat anunexpectedly ighrate (cf.Leblond and Stevens,1948;Storey ndLeblond,1951;F. D. Bertalanffyand Leblond,1953;Leblond and Walker,1956; atableofthe ratesofcell renewal s foundby theLeblond school is given in Bertalanffy,957).The isotope nd other echniques aveshown hattheanimalorganismmaintainstselfna so-calleddynamicor steady state (Schoenheimer,947),chemicalcomponentss well as cells beingcon-tinuallywornout or degraded, nd on the otherhand being replacedby way of resynthesisndthe formationfnew cells. So faras the rate ofcatabolism s concerned,we may assume, as afirst approximation nd based upon various

physiological acts (cf. Bertalanify, 951a), thatit is directly roportional oweight.On theotherhand, mathematical onsiderationsBertalanffy,1941b) show that our basic equation is ratherinsensitive o smallerdeviations f the exponentn from nity.So we mayput,without ny con-siderable oss ofgenerality,heexponent equalto 1.This makesthe olution fourbasicequationmuch asier.The solution f quation5 n = 1) s Bertalanffy,1941b):W = {87/K - [vq/K Wo (1-m)]e-(1_-tn)Kt}1-m (6)withWo = weight t timet = 0.The case is somewhat ifferentithrespect oanabolism.The synthesis f high-molecularellcomponentsneeds, on the one hand, buildingblocks such as amino acids, sugars,phosphates,and so forth,nd on the otherhandenergywhich,in aerobic animals, is provided by oxidativeprocesses.Both can be taken into account aslimiting actors. he experimentalesultsndicatethat, o far s highernimals re concerned,hereis a lawful onnection etween espiration,nab-olism, nd growthwhichworksout in the fol-lowingway.

The exponentn in our basic equationdenotesthedependence f anabolismon bodyweight. fwe insert orm that valuewhich s experimentallyfoundfor the size dependence frestingmetabo-lism, hegrowthaws for heorganismnquestionfollowautomatically.Thus we can predictthegrowthtype of an animal from ts metabolictype, nd thisprediction asprovedto be correctin a largenumber fcases,often n a quiteunex-pectedway.In a first ype, espirations proportionalo the23 power of weight,accordingto the surfacerule.Accordingly,he aw ofgrowth ssumesthefollowingorm:

dW/dt- W2 -KW. (7)We shall not bother with the mathematicalelaboration,but show immediately he results.Fig. 3 givesmetabolismnd growthn the smallaquarium ish, ebistes eticulatus. etabolicrates,measured s oxygenconsumption,re presentedin the log-logor allometric lot. As will be re-membered,n the case of the surfacerule theallometric egressionine has a slope of 23. Sofaras thegrowth urves re concerned,hesolu-tion of the growthequation gives theoretical



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QUANTITATIVE LAWS IN METABOLISM AND GROWTH 225curveswith the followingmain characteristics.First, growthrates are decreasing nd growtheventually ttains a steady state. Secondly,thecurvesforweightgrowth nd linear growth recharacteristicallyifferent.he curve of weightgrowth s sigmoid,with a point of inflexiontabout one-third f the finalweight.The curve oflineargrowth s a decaying xponentialwithoutturningpoint. This is what we actually findexperimentally.This is themost ommon orm fgrowth urves,found n fish,n a number f nvertebratelassesand also, with certainrestrictions,n mammals.The validityof these growth quationshas beenshown n many examples Putter, 1920; Berta-lanffy, 934, 1951a), and theyhave beenadoptedin applied biology.It appears that the "Bertalanffy rowth qua-tion" swidely pplied ninternationalisheries.thas been found o fitthe commerciallyxploitedfish pecies tudiedbythe Fisheries aboratory ftheMinistryfAgriculture,isheries nd Food atLowestoft England), with hepossible exceptionof hehake Wimpenny,ers. ommun.). compre-hensive heoreticalmodel of the dynamics f ex-ploited ish opulations as beendeveloped,where-in growth fthe speciesconcerned s representedby equation 7 (Beverton, 1954; BevertonandHolt, 1957). Discussionofthispopulationmodel(which, partfrom isheries, aywellbe adaptableto otherpopulations) s beyond he scope ofthepresent eview. t shouldbe mentioned, owever,that examination fthe variousgrowth unctionsproposed ed to the conclusion hat "von Ber-talanify's rowth quation s themost atisfactoryofanythat have hitherto eendeveloped" Bever-tonandHolt, .c.). Ampledata as well as descrip-tionofmathematicalnalysis an be foundn thiswork.A relation imilar o thatstated by Bertalanffyet al. forthe surfacedependence f respirationwas foundby Yoshida (1956) in food intake.The quantity fplankton onsumed ythe ardineis proportionalo thesquareofbody ength, ndthe same appears to be true for assimilatingorgans, uchas thegill-rakersnd thegut.This characteristicourse of growth s easilyunderstood.f a body,withnot toomuchchangeof shape, increases n size, its surfaces ncreaseapproximatelywith the second power of thelength,but its volumeand mass with the thirdpower. Hence, the ratio between surface and

200 __ _ - - -

a) C',j -00~o 4-40 ---- - ---30 -e002 0 - -30 40 60 80 100 200 300 500 800Weight n mg.

4 0 - - - - - -- 160

3 5 - -- 14030 -- 6- - 120E Eb) 125----_- 00

I 0 4~~~~-0' 2 0 -

0 -00 2 4 6 8 10 12Time in weeksFIG. 3. IHE FIRST METABOLIC AND GROWTH TYPEMetabolic ate a) and growtlhurvesb) in theGuppy Lebistes eticutatus).rowth urvesfor, :length rowth; - - - weight rowth; alculatedaccordingoequation 7). After ertalanffyndMill-ler 1943).

weight s continually hifted n disfavor f thesurface.Consequently, o long as the animal issmall, surface-proportionalnabolism prevailsover weight-proportionalatabolism, and theanimalgrows.The larger t grows, hemore thesurplus remainingfor growth decreases, andeventually steadystatewill be reached whereanabolism and catabolism balance each other,andgrowthomesto an end.Now we come to the secondtype.We havesaid that in certain animals, for example,ininsects, respiration oes not followthe surfacerule but rather s proportional o weight tself.Let us seewhathappens nthiscase (Fig. 4). Thelog-log lotof metabolic ateagainstbody weightwillgive a linewitha slopeof45?. On theotherhand,we have to insert1 fortheexponentm in



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226 THE QUARTERLY REVIEW OF BIOLOGY8060 - -

a)~~ 0 -j -00 10 -A

5---20 30 40 60 80 100 200 300Weight n mg.(Tenebrio)

5.4- -E1: -?- >

b 2--f

o 20 40 60 80Timqe n hours(Drosophia)FIG. 4. THE SECOND ETABOLICANDGROWnTYasPEMetabolic ate a) andgrowthb) culrveexponent-ial) in insect arvae.AfterBertalanifynd Muller(1943).the growth quation,and thereupon et a com-pletely ifferentrowthurve nda secondgrowthtype. n contrast o the first ype, nabolism ndcatabolism,both being weight-proportional,unat thesame pace. The more atabolismncreases,the more does anabolism. Therefore,growthrateswill not decreasebut always increase, ndthe arger he animalbecomes, hefastertgrows.Growth s not limitedbut exponential, nd nosteadystate s reached.This seemsto be a para-doxricaltateof ffairs,ut s exactlywhathappenls.

Of course, n insect arva does not grow to anyindefiniteize. However,growth s stopped hereby an altogetherdifferentmechanism.It ismetamorphosiswhich abruptly interceptstheexponentialncrease, ven causing a decrease nbody weight s large mounts f tissue re brokendown in order to develop the imago. The samemetabolic nd growth ype also applies to hemi-metabolous nsects ike the walking-stick, herethere is no apparent metamorphosis, ut thehormonalmechanismsresponsiblefor develop-ment appear to be similar.Again, n land snails,which lso belong othistype, xponential rowthis interceptedyseasonalcycles.Finally, we have described a thirdmetabolictype, one wheremetabolic rate is intermediatebetween urface nd weightproportionality,ndwhich s exemplifiedy pondsnails.Againwe cancalculatewhat growth urves re theoreticallyobe expected.f we insert value 23 < m < 1 intothe basic equation, it appears that the growthshouldfollow thirdtype.The curveof weightgrowthdoes not differ ery muchfromthat in

20

0 8jII7a) 6 =30 40 60 80 100 200 300 500 700Weight in mg.

E8 = '- --V4 {>

0 _ _m e0

0 4 8 12 16 20 24Time in weeksFIG. 5. THE THIRD METABOLIC AND GROWTH TYPEMetabolic ate a) and linear rowthb) (diameterofshell) n theramshornnail Planorbisp.). AfterBertalanffyndMuller1943).



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QUANTITATIVE LAWS IN METABOLISM AND GROWTH 227TABLE 6Metabolic ypes ndgrowthypes

Metabolic ype Growthype ExamplesI. Respiration urface-proportional (a) Linear growth urve: attainingwithout Lamellibranchs, ish,inflexion steady state. mammals(b) Weight growth urve: sigmoid, ttain-ing,with inflexion t ca. 13 of finalweight, steadystate.II. Respirationweight-proportional Linear and weightgrowth urves exponen- Insect larvae,Orthoptera,tial, no steadystateattained,butgrowth Helicidaeintercepted y metamorphosisrseasonalcycles.III. Respiration ntermediateetween (a) Linear growth urve:attainingwith n- Planorbidaesurface-and weight-propor- flexion steadystate.tionality (b) Weight growth urve: sigmoid, imilar

to I(b).thefirst ype.Linear growth, owever,s different,as its curve s S-shapedwith an inflexion. gain,our predictions verifiedFig. 5).So in differentnimal classes there redifferentmetabolictypes and differentypes of growth,agreeingwith theoretical xpectation.Table 6givesa survey f examples nvestigated,nd maybe considereds a first raft or new chapternphysiology, amely,a comparativehysiologyfgrowth.Fromthe theory f growthust outlined,manyconsequencescan be derivedwhichhave beenverified mpiricallycf. Bertalanffy957). Onlyone further xamplewill be given.We can com-pare the values of thecatabolicconstant whichwere calculatedfrom hegrowth urves,withthevaluesofprotein urnover s directly eterminedbyexperiment.n a number fcases,theconstantshavebeen verified ith degree fcorrespondencewhich s quitestriking,incethe theoreticalmodelis admittedly versimplifiednd, on the otherhand,the errornthephysiologicaleterminationsofprotein urnovers considerable. or example,in 1938the author alculated heturnoverateofprotein from the growthcurve of man. Thevalue found was 1.165 g./kg.body weight/day.Eleven years later, Sprinson and Rittenberg(1949) calculated protein turnoverfromtheirexperiments ithN15, nd found value of1.3g./kg. body weight/day. nlya soundtheory ouldhavepermitteduchprediction.

GROWTH IN MAMMALS AND MANIt is, however, bvious that the theory epre-sentsa firstpproximation,nd thatwithfurther

development evisionsand the introduction fcomplicatingactorswillbe necessary forrecentdiscussionsof the theory,cf. Duspiva, 1955;Harms, 1955;Linzbach,1955; Zeuthen, 955; Ber-talanffy,957).A case in point is growthn mammals.Fromtheviewpoint f metabolism,t maybe said that,in a firstand crude approximation,mammalsappear to belong to our firsttype,where thesurfacerule of metabolism pplies. Indeed, thesurface ulewas first tatedby Rubnerformam-mals, and it was already mentioned hat theclinically mportant ase, the determinationfbasal metabolism n man, applies the surfacerule n thesomewhatmodified orm f the Duboisstandardformula.Correspondingly, ammaliangrowth oughly ollows he characteristicatternof the first rowth ypediscussed.In detail, however, there are complications.Unfortunatelynd somewhat aradoxically,hereare relativelyfew good data suitable for thistypeofanalysis.If the basal or restingmetabolismn the rat(Fig. 6) is measured ver the entire ife span, itappears thatas a crudeoverall pproximationhesurface uleobtains, hat s, theoverall llometricregressioninehas a slope of about 23. In moredetail,however, here s a break n this ine,suchthat thefirst artof the allometricine s steeper,and the second part much flatter, han wouldcorrespondo a slopeof23. The breaktakesplaceat a bodyweightof about 100 g., that is, pre-ceding exualmaturation.As we have found inother nvestigationsBertalanifynd Pirozynski,1952, 1953), similarbreaks appear in quite a



13/16

228 THE QUARTERLY REVIEW OF BIOLOGY1500 _ .1000 _ \P800 _600

E 400 *thmuSt1n rg.300

CP 1008e0 ib. verQor2 n0t isn50 54 0 430 11II 1II1II1 I I I1 15 20 30 40 60 100 200 400~

Body weight 'n g.FIG. 6. DISCONTINUITIES or RELATivE GRowTHIIN THlE ALBiNo RAT!FAII iscontinuitiesppear t a bodyweight fap-proximately00g., .e.,before uberty. correspond-ingdiscontinuitys foundn thegrowthf the ntireanimalsee Fig. 7). Only egressionines reshownnthefigure;ndividualata and statisticalnalysis regiven n the original apers.After ertalanffyndPiozynski1952, 953) ndRacine 1953).

numberof physiological haracteristics, t thesame age and bodysize: in theallometric rowthof the liver,the involution f the thymus, hetissue espirationf iver ndthymus,ndcertainlythere re others Fig. 6). There is small wonderthat these breaks and shifts re found,as thecoming nto play of the sex hormones ntails adeep-reaching hange in the entire metabolicpattern.On the otherhand, the growth urve of therat was analyzed long beforethe physiologicalstudies ust mentionedBertalanify,938, 1951a;a discussion f recent iterature n rat growthsgiven in Bertalanffy, 957). The result wasfound hatthegrowth f therat follows hefirstgrowth ype,witha characteristichange,how-ever, n thevalues of theconstants, gain at thecriticalpoint of around 100 g. body weight. fthese growth ycles are taken into account,anexcellentfitof the empiricalgrowth urvesbymeansofour formulas an be obtained Fig. 7).So it wouldappearthatmammals elong o thefirst ype,with the qualification, owever, hattwo growth cycles must be distinguished, hetransitionrom he first o the second ycletakingplace at thetimeofsexualmaturation. his is tobe considered o more hana firstpproximation.However,there s one organismwhosegrowth

curve s differentrom ll others. inceourgrowthformulas pply to a large numberof species,the shapes of theirgrowth urves are the same,and the samecurve can be used to representhegrowth f variousspecies, simplyby takingdif-ferentcales for imeand bodysize. Fig. 8 showsthe growth urves of a fishand a mouse. Thelatter,similarto that of the rat, also shows agrowth ycle n detailed analysis,whichdoes notmuchalter the picture. f, however, he growthcurveof man is entered,t appearsto be unique.The secondpart of the curve, beginningwithpuberty,follows the general pattern.The firstpart, however, s verydifferent.n infancy ndchildhood,the curve is enormously rotracted.A new growth ycle s added, as it were,to thetypical patternof growth.Although hischangeis heralded n thegrowth ycles f owermammals,only n mandoes t lead toa singular hapeofthegrowth urve.Thisgrowth urve fman, bnormalas it were, s of courseconnectedwithchanges nthe hormonalbalance. This is demonstrated ypathological cases, such as pubertaspraecox npituitary ysfunction, hen puberty akes placeat an early ge, as it does in othermammals ndstill napes. The singular rowth urve fman s aquantitative expressionof the retardationofhuman developmentwhich, according to Bolk(1926), s one of thebasic factors n the evolutionof man. At the same time, t is an importantfactorfor his uniqueness n nature. Animalsrunthroughheirdevelopmental eriod peedily, nd

300 _ _ _ _ __ 300

E _- - _-Ec200 _ - -_ _ -200

R100 too______

0 100 200 300 400Time in daysFIG. 7. GROWTH OF THE ALBINO RAT

Donaldson's data (d1). Length growthweight growth ---, calculated accordingto equa-tion7).Thecalculationhows wo rowthycles, itha break t approximately00g. bodyweight. onald-son'sdata are oday ot onsideredo be optimal,incemodernaboratoryietshave ncreasedhegrowthnthe rat.Thefigurehows, owever,he excellent u-merical it hatcan be obtainedwith he theoreticalformulas. fter ertalanffy1941b).



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QUANTITATIVE LAWS IN METABOLISM AND GROWTH 229100

_80 - - z-______80~~~~~~~~~~- ________x ~3: (;0 _ _ < Man*c o 5 lo years40 _ 37 Mouse x40 _ _ 5 o0 5 lOweeksAbramis bramao20 0- ~~~0 5 0

20 7V _____ years______Time in daysFIG.8. CO1PARISON0FHE GROWTH CURVESOF A FISH,MOUSE, ND MAN

After ertalanffy1951a).soon they reach pubertyand the adult stage.Man, on the otherhand, s given long periodofyouth, nd is thus enabled to learn and to collectexperience. hus the characteristic umangrowthcurve is a prerequisite or that mental develop-mentand civilizationwhich so sharply distin-guishesman from ll otherbeings.

ACKNOWLEDGMENTThis ssay s baseduponwork arriedhroughn theauthor'saboratoriest the Universityf Vienna ndtheUniversityfOttawa Canada).Part of thisworkwas aidedby research rants rom he NationalRe-

search ouncil,heNational ancernstitute,ndtheHumanitiesesearch ouncil fCanada.A surveyfthe problemfanimalgrowthn generals given nBertalanffy,951a, nd 1957.SUMMARY

Work imed t establishingonnectionsetweenmetabolismnd growths reviewed.n thevariousanimal classes, three "metabolic types," i.e.,forms f dependence f metabolicrate on bodysize can be distinguished:proportionality fmetabolicrate to surface rea, or to weight, rone intermediate etween surface and weightproportionality.he various theoriesregardingthe size dependence fmetabolismre discussed,withparticular onsideration f the relationoftissuerespirationo body size. Correspondingothe "metabolic types" mentioned,there are"growth ypes"distinguishedy differentrowthcurvesofthespeciesconcerned.A general heoryof nimalgrowth,dvancedbythe uthor, ermitsexplanation f the connection etweenmetabolictypes nd growth ypes.The theorys illustratedby examplestakenfromnvertebratend verte-brate classes. Mammalian and human growth,whilegenerally ollowinghe "first rowth ype,"showbreaks n thegrowth urveconnectedwithpuberty. he data and concepts resented eralda new chapterof physiology, he comparativephysiologyfgrowth.

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. 1940. Untersuchungeniber ie Gesetzlichkeitdes Wachstums. III. QuantitativeBeziehungenzwischen Darmoberflache nd KBrpergrosse eiPlanaria maculata. Arch. EntwickMeck.Org.,140: 81-89.

. 1941a. Untersuchungen ber die Gesetzlich-keit des Wachstums. IV. Probleme einerdyna-mischenMorphologie. Biol. gen., 15: 1-22.. 1941b. Untersuchungeniber die Gesetzlich-keit des Wachstums. VII. Stoffwechseltypenund Wachstumstypen. Biol. Zbl., 61: 510-532.. 1942. Untersuchungenberdie Gesetzlichkeitdes Wachstums. V. Wachstumsgradientenndmetabolische Gradienten bei Planarien. Biol.Gen., 15: 295-311.. 1942, 1951a. Theoretische iologie. Bd. ILStoffwechsel,Wachstum. 1. Aufl. Borntraeger,Berlin. 2. Aufl. Francke,Bern.. 1948. Das organische Wachstum und seineGesetzmassigkeiten.Experientia,4: 255-269.. 1949. Problems of organic growth. Nature,Lond., 163: 156-158.. 1951b. Metabolic types and growth types.Amer. Nat., 85: 111-117.. 1953. Biophysikdes Fliessgleichgewiclsts. -bers.vonW. H. Westphal. Vieweg,Braunschweig.



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230 THE QUARTERLY REVIEW OF BIOLOGYBERTALANFFY, L. VON. 1957. Wackstum. Kuiken-thal's Handbuch der Zoologie. Vol. VIII 4 (6).de Gruyter, erlin.- , andR. R. ESTWICK. 1953. Tissue respiration fmusculature n relation to body size. Amer. J.

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Bertalanffy Ludwig Von 1957 Quantitative Laws in Metabolism and Growth

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